Reasoning About Division

Use place value understanding and properties of operations to perform multi-digit arithmetic

6

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

GRAPHICS ON SCREEN
Reasoning About Division
Lynn Simpson OC So here's the first problem. And I want you to think about this. And as always give me a thumb when you think you have an answer.
Lynn Simpson VO/OC My name is Lynn Simpson.
GRAPHICS ON SCREEN
Lynn Simpson 4th Grade Lakeridge Elementary, Renton, WA
And I'm a fourth grade teacher at Lakeridge Elementary in the Renton School District.
Lynn Simpson OC Say it in your hand. Let it go.
Group OC 100.
Lynn Simpson VO Who would like to tell me about how they knew it was 100? Nashun.
Nashun VO Because I know that you can use multiplication to help you and that I know that three times 100 equals 300.
Lynn Simpson VO Today we're going to give the kids a division problem and we call it a number string. And the purpose of this number string is to have them use what they know about place value to divide a larger number. We're going to divide 300 by three. Then we're going to divide 120 by three and use what we...
GRAPHICS ON SCREEN
Common Core: Find Whole Number Quotients Using Strategies Based On Place Value
...know about those place values and how those go together to then divide 420 by three.
Lynn Simpson VO I think some of you probably already know what I'm going to do next, but let's find out.
Female Student #1 VO Divided by three equals.
Female Student #2 VO Equals. Oh, and then four.
Lynn Simpson VO Turn and talk in just a second about what you think it is and why you think it is that way, okay? Ready? Go.
Lynn Simpson VO/OC What I'm looking for is if they can articulate where those two parts are in the whole of the larger problem. And why it is that we can do that in two separate chunks.
Lynn Simpson OC Can you explain why you think it's 140?
Female Student In Pink Top OC You basically have to add the first two numbers and then you could just add the first two answers.
Lynn Simpson OC So because I added these two dividends, I could add.
Female Student In Pink Top VO Those.
Lynn Simpson OC The answers right here.
Female Student In Pink Top VO Yeah.
Lynn Simpson OC To get that?
Female Student In Pink Top OC Yes.
Lynn Simpson VO/OC Okay. They do try to explain it in different ways. Because if their first way didn't work, they've got to think of a different way to explain it. So they become teachers to each other.
Female Student In Red Top OC/VO Because 300 divided by three equals 100. And 120 divided by three equals 40. We just add the answers.
Lynn Simpson VO/OC So I'm teaching them how to think. I'm teaching them how to explain their way of seeing this problem. And if they can all understand what they're all saying and realize that you can model this in fractions, you can model it with an array or you can do computation, and you all end up with the same understanding, that's like three times as good as having just the right answer.
Lynn Simpson OC True or false?
Lynn Simpson VO The kids love true false because they've gotten to the point where they like to have debate.
Lynn Simpson OC Now remember...
GRAPHICS ON SCREEN
Common Core: Make Sense Of Problems And Persevere While Solving Them
...if you think it's true or if you think it's false, you want to have a really good reason, okay.
Lynn Simpson VO/OC There are times when paper and pencil come out. Where drawings get made, examples get made. They model. They'll pull out manipulatives. They'll say, "Look, if we do this." So what they're trying to do is help the other person understand their way of seeing it. And then the person they're talking to, if they're not convinced, will do the same thing back.
Lynn Simpson OC/VO 80 divided by four is the same as 80 divided by two and 80 divided by two. I could put a story problem around this. 80 Skittles divided by four kids. It's the same amount as 80 Skittles divided by two kids plus 80 Skittles divided by two kids. I want you to think carefully. Remember, you need to have a reason that you think the way you do. Go ahead.
Group OC I think it's false.
Male Student In Red Shirt OC I think it's false.
Male Student In Red & White Jacket VO/OC It's true because if you switch it around, 80 divided by two and then <inaudible> 80 divided by two will equal 80 divided by four.
Lynn Simpson VO/OC It's very evident that the rigor of learning for the kids has increased. And it has also increased the rigor of teaching. But they are responding to that and they're being successful with that. And then the benefit of that for them is that they feel like they can do anything.
Lynn Simpson OC Show me a thumb if you think it is true.
GRAPHICS ON SCREEN
Common Core: Students Construct Viable Arguments And Critique The Reasoning Of Others
Show me a thumb if you think it is false. So we have some people that think both ways. So let's take some people who want to support false. Let's start with Jason. Would you defend true?
Jason OC Because you're cutting in half four. So it's 80 divided by two. And then it's another 80 divided by two because two plus two equals four.
Lynn Simpson VO/OC Okay. Romy, what are you thinking? You're agreeing.
Romy VO/OC I think it was true because two plus two is four. So then it will be the same thing.
Lynn Simpson VO Okay. Akiah, what are you thinking?
Akiah OC What I'm thinking was that 80 divided by four is 20 and then you have the equal sign. It's 20 is the same as 80 divided by two which is 40 plus 40 which is, it's false. Because 40 plus 40 is not 20. It's 80.
Lynn Simpson OC Go ahead, Biffy.
Biffy OC I think it's false because if you do 80 divided by two, then you already split all the Skittles so you can't do it again because all the Skittles are already gone. You can't change the divisor. You can only change the dividend.
Lynn Simpson OC/VO So what do you think, Jason, about what she said? She said if you split 80 Skittles between two people, you've used them up and you can't do it again.
Jason OC I think she might be right. So I'm starting to think it's false.
Lynn Simpson VO So you're starting to wonder? Okay.
Lynn Simpson VO/OC The increased rigor of the Common Core and of what we're expecting them to do is a matter of practice. They're more than happy to explain what they think to anybody who asks. And that sort of confidence along with the content knowledge that they're learning is preparing them for middle school and high school in a way that we have never seen before.
Lynn Simpson VO Did you revise your thinking? Did we help you revise your thinking? Right.
Lynn Simpson VO/OC It makes me want to come to work when the kids are that encouraged and that excited about learning. They realized finally how hungry they are to know more. And that they can make sense of it is the most empowering thing in the world.
Lynn Simpson OC/VO Clap once if you can hear my voice. Clap two times. Give me a hoorah.
Group OC Hoorah.
Lynn Simpson OC Good.